I'm pretty sure the dog that Anastasia was holding on a leash on the Standart was not Shvibzik. Shvibzik died of cerebral inflammation in mid 1915. Which was about the same time as when the picture of Anastasia in the motorcar with a dog was taken. Shvibzik was also a type of pomerianian dog which is what the "teddy" is. The dog in the Standart picture is clearly not.
There is no proof that Jemmy was killed IN the cellar with the family and Yurovsky gives no mention of any dogs present at the actual executions, however, I remember reading that Jemmy was making so much noise with her barking and everything that they did kill her. Her body was found frozen in a mineshaft on June 25, 1919, very well preserved and was positively identified as Jemmy by Pierre Gilliard. When found her, forensic pathologists did an autopsy which showed that the dog had been dead for no more than a week when it was discovered. So, it couldn't have died with Anastasia. Jemmy was given to Tatiana as a gift from Anya Vyrubova after she left the family. Since Anastasia did not have a dog but loved them very much, she played with Jemmy all the time and kind of took her under her wing. Her legs were so short that she had to carry him up the stairs and Anastasia often referred to her as "My Jim." But it still was not technically hers. I read that Ortino was killed on the staircase and from the extra bone fragments, is believed to have also been burned along with the others. Joy survived the Ipatiev House and lived out the remainder of his life at Windsor, where he is now buried.